Disks, Washers and Shells 6.2 Volumes Disks, Washers and Shells LIMERICK GENERATING STATION Limerick Generating Station, located in Limerick Township, Montgomery County, PA, is a two-unit nuclear generation facility capable of producing enough electricity for over 1 million homes. The plant site is punctuated by two natural-draft hyperbolic cooling towers, each 507 feet tall, which help cool the plant. Limerick's two boiling water reactors, designed by General Electric, are each capable of producing 1,143 net megawatts. Unit 1 began commercial operation in February 1986, with Unit 2 going on-line in January 1990. Limerick Nuclear Generating Station, Pottstown, Pennsylvania Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2003

Suppose I start with this curve. My boss at the ACME Rocket Company has assigned me to build a nose cone in this shape. So I put a piece of wood in a lathe and turn it to a shape to match the curve.

r= the y value of the function How could we find the volume of the cone? One way would be to cut it into a series of thin slices (flat cylinders) and add their volumes. The volume of each flat cylinder (disk) is: In this case: r= the y value of the function thickness = a small change in x = dx

The volume of each flat cylinder (disk) is: If we add the volumes, we get:

This application of the method of slicing is called the disk method This application of the method of slicing is called the disk method. The shape of the slice is a disk, so we use the formula for the area of a circle to find the volume of the disk. If the shape is rotated about the x-axis, then the formula is: A shape rotated about the y-axis would be:

y-axis is revolved about the y-axis. Find the volume. The region between the curve , and the y-axis is revolved about the y-axis. Find the volume. We use a horizontal disk. y x The thickness is dy. The radius is the x value of the function . volume of disk

The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of this equation revolved about the y-axis: The volume can be calculated using the disk method with a horizontal disk.

and is revolved about the y-axis. Find the volume. The region bounded by and is revolved about the y-axis. Find the volume. If we use a horizontal slice: The “disk” now has a hole in it, making it a “washer”. The volume of the washer is: outer radius inner radius

This application of the method of slicing is called the washer method This application of the method of slicing is called the washer method. The shape of the slice is a circle with a hole in it, so we subtract the area of the inner circle from the area of the outer circle. The washer method formula is:

If the same region is rotated about the line x=2: The outer radius is: The inner radius is: r R

We can use the washer method if we split it into two parts: Find the volume of the region bounded by , , and revolved about the y-axis. We can use the washer method if we split it into two parts: inner radius cylinder outer radius thickness of slice

Here is another way we could approach this problem: cross section If we take a vertical slice and revolve it about the y-axis we get a cylinder. If we add all of the cylinders together, we can reconstruct the original object.

r is the x value of the function. h is the y value of the function. cross section The volume of a thin, hollow cylinder is given by: r is the x value of the function. h is the y value of the function. thickness is dx.

This is called the shell method because we use cylindrical shells. cross section If we add all the cylinders from the smallest to the largest:

Find the volume generated when this shape is revolved about the y axis. We can’t solve for x, so we can’t use a horizontal slice directly.

If we take a vertical slice and revolve it about the y-axis we get a cylinder. Shell method: This model of the shell method and other calculus models are available from: Foster Manufacturing Company, 1504 Armstrong Drive, Plano, Texas 75074-6027 Phone/FAX: (972) 424-3644 http://home.flash.net/~fmco

Note: When entering this into the calculator, be sure to enter the multiplication symbol before the parenthesis.

When the strip is parallel to the axis of rotation, use the shell method. When the strip is perpendicular to the axis of rotation, use the washer method. p